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We report many-body calculations of the self-energy and lifetime of Shockleyand image states on the (100) and (111) surfaces of Cu that go beyond the $GW$approximation of many-body theory. The self-energy is computed in the frameworkof the GW\Gamma approximation by including short-range exchange-correlation(XC) effects both in the screened interaction W (beyond the random-phaseapproximation) and in the expansion of the self-energy in terms of W (beyondthe GW approximation). Exchange-correlation effects are described withintime-dependent density-functional theory from the knowledge of an adiabaticnonlocal XC kernel that goes beyond the local-density approximation.

Self-energy and lifetime of Shockley and image states on Cu(100) and Cu(111): Beyond the GW approximation of many-body theory